Third, I know that, if
$$Z_t = \cos(\lambda \mathscr{A}_t)$$
$$W_t = -\frac{f'(t)}{2} (X_t^2 + Y_t^2) + f(t)$$
and if $f(t) = -\log\cosh(\lambda(r-t))$, then $Z_t \exp(W_t)$ is a martingale. I also know what the canonical decompositions of these processes are and that $\langle Z, W \rangle_t = 0$. (See this question.)

The fact that $Z_t \exp(W_t)$ allows me to compute $E[Z_t \exp(W_t)] = E[Z_0 \exp(W_0)] = \frac{1}{\cosh(\lambda r)}$. This should be relevant but I don't know how to relate this back to $E[\cos(\lambda \mathscr{A}_t)] = E[Z_t]$.